Eccentricity (mathematics)

All types of conic sections, arranged with increasing eccentricity. Note that curvature decreases with eccentricity, and that none of these curves intersect.

In mathematics, the eccentricity, denoted e or ε{\displaystyle \varepsilon }, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as e.

The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis vertical, the eccentricity is[1]

where β is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and the horizontal. For β=0{\displaystyle \beta =0} the plane section is a circle, for β=α{\displaystyle \beta =\alpha } a parabola. (The plane must not meet the vertex of the cone.)

The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axisa: that is, e=ca{\displaystyle e={\frac {c}{a}}}. (Lacking a center, the linear eccentricity for parabolas is not defined.)

The eccentricity is sometimes called the first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called the numerical eccentricity.

In the case of ellipses and hyperbolas the linear eccentricity is sometimes called the half-focal separation.

The eccentricity of an ellipse is strictly less than 1. When circles (which have eccentricity 0) are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from the category of ellipses, then the eccentricity of an ellipse is strictly greater than 0.

The eccentricity of an ellipse is, most simply, the ratio of the distance f between the center of the ellipse and each focus to the length of the semimajor axis a.

e=fa.{\displaystyle e={\frac {f}{a}}.}

The eccentricity is also the ratio of the semimajor axis a to the distance d from the center to the directrix:

e=ad.{\displaystyle e={\frac {a}{d}}.}

The eccentricity can be expressed in terms of the flatteningg (defined as g = 1 – b/a for semimajor axis a and semiminor axis b):

e=g(2−g).{\displaystyle e={\sqrt {g(2-g)}}.}

(Flattening is denoted by f in some subject areas, particularly geodesy.)

Define the maximum and minimum radii rmax{\displaystyle r_{\text{max}}} and rmin{\displaystyle r_{\text{min}}} as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semimajor axis a, the eccentricity is given by

Ellipses, hyperbolas with all possible eccentricities from zero to infinity and a parabola on one cubic surface.

The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane). But: conic sections may occur on surfaces of higher order, too (see image).

In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocenter distance is close to the pericenter distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipses, in Keplerian, i.e., 1/r{\displaystyle 1/r} potentials.

The eccentricity is also a concept which is used to characterize statistical distribution of data points around a common axis. For example, the eccentricity can be used to characterize shapes of jets of many particles.[4]

The definition closely follows the original "geometrical" concept, with one important difference – data points can have "weights". Such weights can lead to a deviation from the standard geometrical concept that assumes that all data points have the same contributions.